A Novel Framework for Parameter and State Estimation of Multicellular Systems Using Gaussian Mixture Approximations

Abstract

Multicellular systems play an important role in many biotechnological processes. Typically, these exhibit cell-to-cell variability, which has to be monitored closely for process control and optimization. However, some properties may not be measurable due to technical and financial restrictions. To improve the monitoring, model-based online estimators can be designed for their reconstruction. The multicellular dynamics is accounted for in the framework of population balance models (PBMs). These models are based on single cell kinetics, and each cellular state translates directly into an additional dimension of the obtained partial differential equations. As multicellular dynamics often require detailed single cell models and feature a high number of cellular components, the resulting population balance equations are often high-dimensional. Therefore, established state estimation concepts for PBMs based on discrete grids are not recommended due to the large computational effort. In this contribution a novel approach is proposed, which is based on the approximation of the underlying number density functions as the weighted sum of Gaussian distributions. Thus, the distribution is described by the characteristic properties of the individual Gaussians, like the mean and covariance. Thereby, the complex infinite dimensional estimation problem can be reduced to a finite dimension. The characteristic properties are estimated in a recursive approach. The method is evaluated for two academic benchmark examples, and the results indicate its potential for model-based online reconstruction for multicellular systems.

Multi-Rate Data Fusion for State and Parameter Estimation in (Bio-)Chemical Process Engineering

“…For more complex cases with multiple unknown parameters, higher nonlinearities or non-Gaussian process noise, performance of the UKF approach may deteriorate. Here, particle filters [6,32] can represent a promising alternative. Furthermore, the outlined approach could be used to extend existing estimators for distributed parameter systems as found in the description of particle formation [33][34][35] and biotechnological processes [36][37][38], to reconstruct the systems states and parameters in presence of measurement delays or multi-rate measurements.…”

Special Issue on “Recent Advances in Population Balance Modeling”

“…They used parametric bootstrap to illustrate confidence intervals and identifiability of the parameters. Dürr and Waldherr [8] look into the challenge with lack of data availability in multi-cellular systems where some properties may be impossible to measure due to economic or operational constraints. Therefore, they proposed an approach based on approximation of the underlying number density functions as the weighted sum of Gaussian distributions.…”

Stochastic Models to Qualify Stem Tapers

“…In this study, newly developed stem tapers are defined (as segmented models) by using one or two joining points (a 0 and a 1 ) to weld three stochastic processes, defined by Equations (1), (5), and (9). In the sequel, the fixed effects parameters for the bottom part of a stem are listed by index B, the middle part by index M, and for the top part, they are listed by index T. The stem taper SDE models with two joining points a 0 and a 1 are defined in the following forms, respectively:…”

“…Fundamental SDE theory is defined on random variables. The universality of random processes accounts for the wide range of applications of the theory, including human population [2], forestry [3,4], biology [5], and epidemiology [6]. In biological systems, SDEs are used in place of deterministic models, obtained by including a noise term in the ordinary differential equation of the respective deterministic model [7].…”